Chapter 4.1, Problem 6C

This phenomenon is so common, in fact, that data sets that follow a similar pattern are said to be normally distributed. Things like sizes of individuals, IQs, weights of packaged products, and lifespans of batteries or lightbulbs are often normally distributed.

I only collected 100 pods, out of thousands (maybe millions) on the tree. If I had collected a lot more of them, the picture would most likely have started to look a lot like the nice, symmetric diagram to the right. When a group of data is normally distributed, and we know the mean and standard deviation, there’s a rule that allows us to estimate how many data values fall within certain ranges. This is known as the empirical rule, and it’s illustrated by the diagram below.

The empirical rule says that when a data set is normally distributed, about 68% of all values will fall within one standard deviation of the mean; about 95% will fall within 2 standard deviations of the mean; and about 99.7% will fall within 3 standard deviations of the mean.

For example, the heights of American men are normally distributed with mean 5 feet 9.3 inches and standard deviation 2.8 inches. That is, $\mu$

is 5 feet 9.3 inches, and $\sigma$ is 2.8 inches.

Use the information just given about $\mu$ and $\sigma$ to fill in the blanks on the next empirical rule diagram with heights, using the formulas below the blanks for guidance.